The $\dot W^{-1,p}$ Neumann problem for higher order elliptic equations

TL;DR

This paper establishes well-posedness and boundary estimates for the Neumann problem of higher order elliptic equations with variable coefficients in the half-space, extending results to negative smoothness boundary data and general p ranges.

Contribution

It introduces new methods to solve the Neumann problem with boundary data in negative smoothness spaces for higher order elliptic equations, expanding known results beyond p=2.

Findings

Well-posedness of Neumann problem for variable coefficient elliptic equations.

Layer potential estimates in L^p and negative smoothness spaces.

Extension of techniques to non-self-adjoint operators.

Abstract

We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in the negative smoothness space $\dot W^{-1,p}$, where $\max(0,\frac{1}{2}-\frac{1}{n}-\varepsilon) <\frac{1}{p} <\frac{1}{2}$. Our arguments are inspired by an argument of Shen and build on known well posedness results in the case $p=2$. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in $L^p$ or $\dot W^{\pm1,p}$ for a similar range of $p$, based on known bounds for $p$ near $2$; in this case we may relax the requirement of self-adjointess.